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% ambfn1.m - plots ambiguity function of a signal u_basic (row vector)%% The m-file returns a plot of quadrants 1 and 2 of the ambiguity function of asignal% The ambiguity function is defined as:%% a(t,f) = abs ( sumi( u(k)*u'(i-t)*exp(j*2*pi*f*i) ) )%% The user is prompted for the signal data:% u_basic is a row complex vector representing amplitude and phase% f_basic is a corresponding frequency coding sequence%% The duration of each element is tb (total duration of the signal is tb*(m_basic-1))%% F is the maximal Dopler shift% T is the maximal Delay% K is the number of positive Doppler shifts (grid points)% N is the number of delay shifts on each side (for a total of 2N+1 points)% The code allows r samples within each bit%% Written by Eli Mozeson and Nadav Levanon, Dept. of EE-Systems, Tel Aviv University

% clear all

% prompt for signal datau_basic=input(' Signal elements (row complex vector, each element last tb sec) =? ');m_basic=length(u_basic);

fcode=input(' Allow frequency coding (yes=1, no=0) = ? ');if fcode==1

f_basic=input(' Frequency coding in units of 1/tb (row vector of same length) = ? ');end

F=input(' Maximal Doppler shift for ambiguity plot [in units of 1/Mtb] (e.g., 1)= ? ');K=input(' Number of Doppler grid points for calculation (e.g., 100) = ? ');df=F/K/m_basic;

T=input(' Maximal Delay for ambiguity plot [in units of Mtb] (e.g., 1)= ? ');N=input(' Number of delay grid points on each side (e.g. 100) = ? ');

sr=input(' Over sampling ratio (>=1) (e.g. 10)= ? ');r=ceil(sr*(N+1)/T/m_basic);

if r==1dt=1;

m=m_basic;uamp=abs(u_basic);phas=uamp*0;phas=angle(u_basic);if fcode==1

phas=phas+2*pi*cumsum(f_basic);enduexp=exp(j*phas);u=uamp.*uexp;

else % i.e., several samples within a bit

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dt=1/r; % interval between samplesud=diag(u_basic);ao=ones(r,m_basic);m=m_basic*r;u_basic=reshape(ao*ud,1,m); % u_basic with each element repeated r timesuamp=abs(u_basic);phas=angle(u_basic);u=u_basic;if fcode==1

ff=diag(f_basic);phas=2*pi*dt*cumsum(reshape(ao*ff,1,m))+phas;uexp=exp(j*phas);u=uamp.*uexp;

endend

t=[0:r*m_basic-1]/r;tscale1=[0 0:r*m_basic-1 r*m_basic-1]/r;dphas=[NaN diff(phas)]*r/2/pi;

% plot the signal parametersfigure(1), clf, hold offsubplot(3,1,1)plot(tscale1,[0 abs(uamp) 0],'linewidth',1.5)

ylabel(' Amplitude ')axis([-inf inf 0 1.2*max(abs(uamp))])

subplot(3,1,2)plot(t, phas,'linewidth',1.5)axis([-inf inf -inf inf])ylabel(' Phase [rad] ')

subplot(3,1,3)plot(t,dphas*ceil(max(t)),'linewidth',1.5)axis([-inf inf -inf inf])xlabel(' \itt / t_b ')ylabel(' \itf * Mt_b ')

% calculate a delay vector with N+1 points that spans from zero delay to ceil(T*t(m))% notice that the delay vector does not have to be equally spaced but must haveall% entries as integer multiples of dt

dtau=ceil(T*m)*dt/N;tau=round([0:1:N]*dtau/dt)*dt;

% calculate K+1 equally spaced grid points of Doppler axis with df spacingf=[0:1:K]*df;

% duplicate Doppler axis to show also negative Dopplers (0 Doppler is calculatedtwice)f=[-fliplr(f) f];

% calculate ambiguity function using sparse matrix manipulations (no loops)

% define a sparse matrix based on the signal samples u1 u2 u3 ... um% with size m+ceil(T*m) by m (notice that u' is the conjugate transpose of u)% where the top part is diagonal (u*) on the diagonal and the bottom part is a zero matrix

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%% [u1* 0 0 0 ... 0 ]% [ 0 u2* 0 0 ... 0 ]% [ 0 0 u3* 0 ... 0 ] m rows% [ . . . ]% [ . . . ]% [ . 0 0 . ... um*]% [ 0 0 ]% [ . . ] N rows% [ 0 0 0 0 ... 0 ]%mat1=spdiags(u',0,m+ceil(T*m),m);

% define a convolution sparse matrix based on the signal samples u1 u2 u3 ... um% where each row is a time(index) shifted versions of u.% each row is shifted tau/dt places from the first row% the minimal shift (first row) is zero% the maximal shift (last row) is ceil(T*m) places% the total number of rows is N+1% number of columns is m+ceil(T*m)

% for example, when tau/dt=[0 2 3 5 6] and N=4%% [u1 u2 u3 u4 ... ... um 0 0 0 0 0 0

]% [ 0 0 u1 u2 u3 u4 ... ... um 0 0 0 0]% [ 0 0 0 u1 u2 u3 u4 ... ... um 0 0 0]% [ 0 0 0 0 0 u1 u2 u3 u4 ... ... um 0]% [ 0 0 0 0 0 0 u1 u2 u3 u4 ... ... um]

% define a row vector with ceil(T*m)+m+ceil(T*m) places by padding u with zeroson both sidesu_padded=[zeros(1,ceil(T*m)),u,zeros(1,ceil(T*m))];

% define column indexing and row indexing vectorscidx=[1:m+ceil(T*m)];ridx=round(tau/dt)';

% define indexing matrix with Nused+1 rows and m+ceil(T*m) columns% where each element is the index of the correct place in the padded version ofuindex = cidx(ones(N+1,1),:) + ridx(:,ones(1,m+ceil(T*m)));

% calculate matrixmat2 = sparse(u_padded(index));

% calculate the ambiguity matrix for positive delays given by%% [u1 u2 u3 u4 ... ... um 0 0 0 0 0 0] [u1* 0 0 0... 0 ]% [ 0 0 u1 u2 u3 u4 ... ... um 0 0 0 0] [ 0 u2* 0 0... 0 ]% [ 0 0 0 u1 u2 u3 u4 ... ... um 0 0 0]*[ 0 0 u3* 0... 0 ]% [ 0 0 0 0 0 u1 u2 u3 u4 ... ... um 0] [ .. . ]

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% [ 0 0 0 0 0 0 u1 u2 u3 u4 ... ... um] [ .. . ]% [ . 0 0 . ...um*]%

[ 0 0 ]%[ . . ]% [ 00 0 0 ... 0 ]

%% where there are m columns and N+1 rows and each element gives an element% of multiplication between u and a time shifted version of u*. each row gives% a different time shift of u* and each column gives a different entry in u.%uu_pos=mat2*mat1;

clear mat2 mat1

% calculate exponent matrix for full calculation of ambiguity function. the exponent% matrix is 2*(K+1) rows by m columns where each row represents a possible Doppler and% each column stands for a differnt place in u.

e=exp(-j*2*pi*f'*t);

% calculate ambiguity function for positive delays by calculating the integral for each% possible delay and Doppler over all entries in u.% a_pos has 2*(K+1) rows (Doppler) and N+1 columns (Delay)a_pos=abs(e*uu_pos');

% normalize ambiguity function to have a maximal value of 1a_pos=a_pos/max(max(a_pos));

% use the symmetry properties of the ambiguity function to transform the negative Doppler

% positive delay part to negative delay, positive Dopplera=[flipud(conj(a_pos(1:K+1,:))) fliplr(a_pos(K+2:2*K+2,:))];

% define new delay and Doppler vectorsdelay=[-fliplr(tau) tau];freq=f(K+2:2*K+2)*ceil(max(t));

% exclude the zero Delay that was taken twicedelay=[delay(1:N) delay((N+2):2*(N+1))];a=a(:,[1:N (N+2):2*(N+1)]);

% plot the ambiguity function and autocorrelation cut[amf amt]=size(a);

% create an all blue color mapcm=zeros(64,3);cm(:,3)=ones(64,1);

figure(2), clf, hold offmesh(delay, [0 freq], [zeros(1,amt);a])

hold onsurface(delay, [0 0], [zeros(1,amt);a(1,:)])

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